Mathematical Conversations

Multicolor Problems, Problems in the Theory of Numbers, and Random Walks
Dover Publications (Verlag)
erschienen am 30. Oktober 2013 | 288 Seiten
E-Book | ePUB mit Adobe DRM | Systemvoraussetzungen
978-0-486-15491-6 (ISBN)
Combining three books into a single volume, this text comprises Multicolor Problems, dealing with several of the classical map-coloring problems; Problems in the Theory of Numbers, an elementary introduction to algebraic number theory; and Random Walks, addressing basic problems in probability theory.The book's primary aim is not so much to impart new information as to teach an active, creative attitude toward mathematics. The sole prerequisites are high-school algebra and (for Multicolor Problems) a familiarity with the methods of mathematical induction. The book is designed for the reader's active participation. The problems are carefully integrated into the text and should be solved in order. Although they are basic, they are by no means elementary. Some sequences of problems are geared toward the mastery of a new method, rather than a definitive result, and others are practice exercises, designed to introduce new concepts. Complete solutions appear at the end.
Dover Publications
New York
Höhe: 216 mm | Breite: 137 mm
19,14 MB
281 gr
978-0-486-15491-6 (9780486154916)
0486154912 (0486154912)
weitere Ausgaben werden ermittelt
E. B. Dynkin and V. A. Uspenskii
  • Cover
  • Title Page
  • Copyright Page
  • Contents
  • Section I: Multicolor Problems
  • Introduction
  • Chapter 1. Coloring with Two Colors
  • 1. Simple Two-Color Problems
  • 2. Problems on Square Boards
  • 3. Problems Involving Even and Odd Numbers
  • 4. Networks and Maps
  • 5. General Two-Color Problems
  • Chapter 2. Coloring with Three Colors
  • 6. A simple Three-Color Problem
  • 7. Problems on Hexagonal Boards
  • 8. Dual Diagrams
  • 9. Triangulation
  • 10. Dual Maps
  • 11. Normal Maps in Three Colors
  • Chapter 3. The Four-Color Problem
  • 12. Normal Maps in Four Colors
  • 13. Volynskii's Theorem
  • Chapter 4. The Five-Color Theorem
  • 14. Euler's Theorem
  • 15. The Five-Color Theorem
  • Concluding Remarks
  • Appendix
  • Coloring a Sphere with Three Colors
  • Solutions to Problems
  • Bibliography
  • Section II: Problems in the Theory of Numbers
  • Chapter 1. The Arithmetic of Residue Classes
  • 1. Arithmetic, Modulo m, or m-Arithmetic
  • 2. Arithmetic, Modulo p, or p-Arithmetic
  • 3. Extraction of Square Roots in Quadratic Equations
  • 4. Extraction of Cube Roots
  • 5. Polynomials and Equations of Higher Degree
  • Chapter 2. m-Adic and p-Adic Numbers
  • 6. The Division of Multidigit Numbers Using 10-Arithmetic
  • 7. Numbers with an Infinite Number of Digits
  • 8. m-Adic Number System
  • 9. m-Adic Numbers
  • 10. p-Adic Numbers
  • 11. Geometric Progressions
  • 12. Extraction of Square Roots in Quadratic Equations
  • Chapter 3. Applications of m-Arithmetic and p-Arithmetic in Number Theory
  • 13. The Fibonacci Sequence
  • 14. Fibonacci Sequences in m-Arithmetic
  • 15. The Distribution of the Numbers Divisible by m in A Fibonacci Sequence
  • 16. The Fibonacci and Geometric Sequences
  • 17. Fp-Sequences
  • 18. Pascal's Triangle
  • 19. Fractional Linear Functions
  • Chapter 4. Further Remarks on the Fibonacci Sequence and Pascal's Triangle
  • 20. The Application of p-Adic Numbers to the Fibonacci Sequence
  • 21. The connection between Pascal's Triangle and the Fibonacci Sequence
  • 22. The Distribution of Numbers Divisible by given Numbers in a Fibonacci Sequence
  • Chapter 5. The Equation x2 2 5y2 5 1
  • 23. Solutions of the Equation
  • Concluding Remarks
  • Solutions to Problems
  • Section III: Random Walks
  • Introduction
  • Chapter 1. Probability
  • 1. Fundamental Properties of Probability
  • 2. Conditional Probability
  • 3. The Formula for Complete Probability
  • Chapter 2. Problems Concerning a Random Walk on an Infinite Line
  • 4. Graph of Coin Tosses
  • 5. The Triangle of Probabilities
  • 6. Central Elements of the Triangle of Probabilities
  • 7. Estimation of Arbitrary Elements of the Triangle
  • 8. The Law of the Square Root of n
  • 9. The Law of Large Numbers
  • Chapter 3. Random Walks with Finitely Many States
  • 10. Random Walks on a Finite Line
  • 11. Random Walks Through a City
  • 12. Markov Chains
  • 13. The Meeting Problem
  • Chapter 4. Random Walks with Infinitely Many States
  • 14. Random Walks on an Infinite Path
  • 15. The Meeting Problem
  • 16. The Infinitely Large City with a Checkerboard Pattern
  • Concluding Remarks
  • Solutions to Problems

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